Finite difference heat transfer analyses in Excel

In this first blog (after the hello world one) I’m tackling something I’ve wanted to do for some time

now. That is setting up and solving a simple heat transfer problem using the finite difference (FDM) in MS Excel. The aim is to solve the steady-state temperature distribution through a rectangular body, by dividing it up into nodes and solving the necessary equations only in two dimensions. I’m

Finite difference analyses (FDA’s) are generally performed to predict the values of physical properties at discrete points throughout a body. In the case of a stationary body where heat transfer is primary phenomena, the temperature could be determined throughout as a function of heating or cooling on the boundaries, and the physical properties (heat transfer coefficient) of the material.

By definition, the FDM refers to a method for the numerical solution of differential equations. The solution to the solving the FDA as a whole that is described here is more methodological, relying rather on Excel’s ability to solve iterative loops, than performing matrix algebra.

Purpose

Now why would I use Excel for FDM’s? There are primarily two reasons, and defines the purpose of this post:

1. Firstly, this helps to illustrates the basic concepts of the FDM, and someone following this should at the end have be able to apply the same principles in any other programming platform, and for problems other than simple heat transfer.

2. Secondly, certain aspects of some problems that extend beyond simple heat transfer are sometimes challenging to solve using available FDM or related software. An example of this could be a system where a material chemically reacts as a function of the composition and temperature in each node, and the chemical reactions need to be modeled using third party software. Although not the most efficient, Excel still remains a fairly flexible platform allowing for integration with many other software components.

Using Excel for to perform a finite difference analysis (FDA) does however have is its downsides. The system will be greatly simplified, and in the simplest form, the domain is only defined with fixed sized nodes being rectangular in shape.

Definition

To start off with the solution, the partial differential equation of the governing phenomena needs to be defined, in this case heat transfer. To arrive at the PDE, Fourier’s law is considered that

defines that the negative of the gradient of temperature and area perpendicular to the direction heat is flowing. In one-dimensional form heat flux through a node is:

where q is the heat flux (W/m2), T is the temperature (Kelvin) at a node through which heat is flowing, and k is the conductivity (W/mK) of the material through which heat is flowing. 1D Heat Transfer

The body for the temperature distribution which needs to be solved is broken up into a number of finite elements, or nodes, as illustrated below. Consider the node numbered x somewhere inside the body. The node to the left is numbered x-1, and to the right is x+1.

Heat is flowing through it in the x direction, and because we are only considering the steady state at the moment, the heat flux over the left boundary should be equal to the heat flux over the right boundary. Or in general, the heat flux into the node should be the heat flux out of the node. The heat balance for the node x can be written as follows, discretized, and reduced to a single equation for the temperature of node x as function of the temperatures to the left and right:

Boundary conditions

The above sections covered nodes inside the body, but the nodes on the boundaries need different equations to be solved. For illustration, the problem was defined to have convective heat transfer (cooling) on the left side with a heat transfer coefficient of 5 W/m2K, and have a fixed temperature of 1600?C on the right side. The temperature equation for first node can be derived defining the heat flux into the first node as a function of the heat transfer coefficient: